Chapter 5
The Time Value of Money
Learning Objectives
§ Explain the mechanics of compounding, which is how money grows over a time when it is invested.
§ Be able to move money through time using time value of money tables, financial calculators, and spreadsheets.
§ Discuss the relationship between compounding and bringing money back to present.
Learning Objectives
§ Define an ordinary annuity and calculate its compound or future value.
§ Differentiate between an ordinary annuity and an annuity due and determine the future and present value of an annuity due.
§ Determine the future or present value of a sum when there are non-annual compounding periods.
Learning Objectives
• Determine the present value of an uneven stream of payments
• Determine the present value of a perpetuity.
• Explain how the international setting complicates the time value of money.
Principles Used in this Chapter
• Principle 2:
– The Time Value of Money – A Dollar Received Today Is Worth More Than a Dollar Received in The Future.
Simple Interest
• Interest is earned on principal.
– $100 invested at 6% per year
• 1styear interest is $6.00
• 2ndyear interest is $6.00
• 3rdyear interest is $6.00
– Total interest earned: $18.00
Compound Interest
• When interest paid on an investment during the first period is added to the principal;
• then,
• during the second period, interest is earned on the new sum.
Compound Interest
• Interest is earned on previously earned interest
– $100 invested at 6% with annual compounding
• 1styear interest is $6.00 Principal is $106.00
• 2ndyear interest is $6.36 Principal is $112.36
• 3rdyear interest is $6.74 Principal is $119.11
– Total interest earned: $19.11
Future Value
- The amount a sum will grow in a certain number of years when compounded at a specific rate.
Future Value
FV
FV1 1 = PV (1 + i)= PV (1 + i)
Where
WhereFVFV1 1 = = the future of the investment at the future of the investment at the end of one year
the end of one year i=
i= the annual interest (or discount) the annual interest (or discount) rate
rate PV =
PV = the present value, or original the present value, or original amount invested at the beginning amount invested at the beginning of the first year
of the first year
Future Value
• What will an investment be worth in 2 years?
$100 invested at 6%
FV2 = PV(1+i)2
= $100 (1+.06)2
= $100 (1.06)2
= $112.36
Future Value
• Future Value can be increased by:
• Increasing number of years of compounding
• Increasing the interest or discount rate
Future Value
What is the future value of $500 invested at 8%
for 7 years? (Assume annual compounding)
FVn = PV (1+i)7
= $857
Future Value Using Calculators
Using any four inputs you will find the 5th. Set to P/YR = 1 and END mode.
INPUTS OUTPUT
N I/YR
PMT PV
FV
10 6
0 179.10
-100
Future Value Using Spreadsheets
????
Present Value
The current value of a future payment PV
PV= = FVFVnn{1/(1+i){1/(1+i)nn}}
Where
WhereFVFVnn= = the future of the investment at the future of the investment at the end of n years
the end of n years n=
n= number of years until payment is number of years until payment is received
received i=
i= the interest ratethe interest rate PV =
PV = the present value of the future sum the present value of the future sum of money
of money
Present Value
• What will be the present value of $500 to be received 10 years from today if the discount rate is 6%?
PV = $500 {1/(1+.06)10}
= $500 (1/1.791)
= $500 (.558)
= $279
Present Value Using Calculators
Using any four inputs you will find the 5th. Set to P/YR = 1 and END mode.
INPUTS OUTPUT
N I/YR PMT
PV
FV 10
6 0 100.00
-55.84
Annuity
• Series of equal dollar payments for a specified number of years.
• Ordinary annuity payments occur at the end of each period
Compound Annuity
• Depositing or investing an equal sum of money at the end of each year for a certain number of years and allowing it to grow.
Compound Annuity
FV5= $500 (1+.06)4 + $500 (1+.06)3 +$500(1+.06)2+ $500 (1+.06) + $500
= $500 (1.262) + $500 (1.191) +
$500 (1.124) + $500 (1.090) + $500
= $631.00 + $595.50 + $562.00 +
$530.00 + $500
= $2,818.50
Illustration of a 5yr $500 Annuity Compounded at 6%
5 500
6% 1 2 3 4
0
500
500 500 500
Future Value of an Annuity Using Calculators
Using any four inputs you will find the 5th. Set to P/YR = 1 and END mode.
INPUTS OUTPUT
N
I/YR PMT
PV 5 FV
6 500
0
-2,818.55
Present Value of an Annuity
• Pensions, insurance obligations, and interest received from bonds are all annuities. These items all have a present value.
• Calculate the present value of an
annuity using the present value of
annuity table.
Annuities Due
• Ordinary annuities in which all payments have been shifted forward by one time period.
Amortized Loans
• Loans paid off in equal installments over time
– Typically Home Mortgages – Auto Loans
Payments and Annuities
• If you want to finance a new machinery with a purchase price of $6,000 at an interest rate of 15% over 4 years, what will your payments be?
Future Value Using Calculators
• Using any four inputs you will find the 5th. Set to P/YR = 1 and END mode.
INPUTS OUTPUT
N
I/YR
PMT PV
FV 4
15 6,000
0
-2,101.59
Amortization of a Loan
• Reducing the balance of a loan via annuity payments is called amortizing.
• A typical amortization schedule looks at payment, interest, principal payment and balance.
Amortization Schedule
1,827.51 274.07
$2,101.58 4
1,827.51 1,589.09
512.49
$2,101.58 3
3,416.60 1,381.82
719.76
$2,101.58 2
$4,798.42
$1,201.58
$900.00
$2,101.58 1
Balance Principal
Interest Annuity
Yr.
Compounding Interest with Non-annual periods
• If using the tables, divide the percentage by the number of compounding periods in a year, and multiply the time periods by the number of compounding periods in a year.
Example:
– 8% a year, with semiannual compounding for 5 years.
• Input 8% / 2 = 4% as interest
• Input N = 5*2 = 10 as number of periods
Perpetuity
• An annuity that continues forever is called perpetuity
• The present value of a perpetuity is PV = PP/i
PV = present value of the perpetuity PP = constant dollar amount
provided by the of perpetuity i = annuity interest (or discount rate)
The Multinational Firm
• Principle 1
The Risk Return Tradeoff – We Won’t Take on Additional Risk Unless We Expect to Be Compensated with Additional Return
– The discount rate is reflected in the rate of inflation.
– Inflation rate outside US difficult to predict
– Inflation rate in Argentina in 1989 was 4,924%, in 1990 dropped to 1,344%, and in 1991 it was only 84%.